On the login page, you may have noticed, and even tried checking, the "Stay Logged In" checkbox.  When you log in with this checkbox checked, you are creating a "persistent" login session between the WP server and your computer/browser.  This allows you to access the WP site from that computer without having to enter your login information again, even if several days have passed since you last accessed the site.  There are a couple of important points to remember about persistent logins:

• If you do not access the site for 7 days, your persistent session will expire.
• You can have persistent sessions on multiple computers/browsers simultaneously.  But any given persistent session is only good for that specific computer/browser.
• The "Logout" link in the navigation banner ends the persistent session for the specific computer/browser you are on when you click that link.
• You can end all of your persistent logins (across all computers/browsers) by logging in and going to Account >> My Profile.  On your profile page you'll see a button that says "Log me out of all other computers".   Technically, this operation will just end the persistent nature everywhere you have logged in.  It DOES NOT force an immediate logout of all those sessions.
• Clearing your cookies will end your persistent login